The advanced Mayan culture developed
thanks to a complex synthesis of different culture streams arising from
the home agricultural base, influenced by cultural values coming from regions
lying out of the territory of Mayan settlement. Its forming falls to the
so-called early phase of the initial period placed between 1500 — 800 BC.
It was spread step-by-step to the regions of Guatemala, south-eastern Mexico,
Belize, Salvador and north-western Honduras. The construction of beautiful
and splendid cathedral cities, fine arts of sculpture and painting, use
of their own hieroglyphic script, success in astronomy, existence of the
literature and the development of handicraft and trade were the outer expression
of this cultural-economic rise.

The results of Mayan observations and
calculations of astronomical phenomena are concentrated in the Dresden
Codex. It is a band of paper 3.5 meter long set up into 39 sheets making
up 78 pages 8.5 x 20.5 cm. The paper was obtained from the bark of wild-growing
species of fig tree. It is supposed that it originates from Yucatan as
a latter transcription of an elder original. It contains calendrical data,
written in the Mayan dating system, concerning astronomical data and the
sky mechanics, and tables of multiple integers that are to be used for
calculations of planetary movement ephemerids and tropical years, next
to the hieroglyphic texts and numerous depicturings of the Mayan gods and
ritual scenes.

The data contained in the Dresden Codex
were studied by many researchers who suspected they contain astronomical
data. M.Meinshausen (1913), C.E.Guthe (1921) and H.Spinden (1930) were
the first who had been interested in the eclipses tables. E.Foerstemann
has drawn our attention to Venus visibility ephemerides tables; he also
issued the Dresden Codex with a commentary in 1892. The analysis of these
ephemerides has been made by J.E.Teeple (1926). R.W.Wilson believed that
some of the data could concern the observations of Mars, Jupiter and Saturn
(1924). The above-mentioned researchers, and lots of others, worked with
the calculation coefficients of 584,283 or 584,285 days accordingly to
Goodman-Martinez-Thompson when converting the Mayan dates into the Christian
dating system, or tried to calculate their own coefficient. For this reason
their conclusions were very diverse.

One of the most important problems
during the studies of various Mayan culture phenomena had been the problem
of correlating the Mayan to our Christian dating system. In present times
we are used to correlate the Mayan dates with the Christian ones using
the Goodman-Martínez-Thompson correlation. Accordingly to it, a
stable coefficient of 584,283 or 584,285 days is added to the Mayan dates
expressing the counts of days which have passed from a particular day to
the date of a certain event. The Mayan date is converted into a Julian
day number and the latter finally to the corresponding day, month and year
of the Julian calendar used in modern astronomy.

Working with the Mayan data of the
Dresden Codex we found that the Goodman-Martínez-Thompson correlation
is unusable, even for the dates evidently concerning certain astronomical
phenomena, such as the observations of Venus visibility, or Sun and Moon
eclipses. We have obtained a new coefficient of 622,261 days for the conversion
of the Mayan dates to our dating system by a complete analysis of the mutual
relations between the time intervals of all the Mayan dates in the Dresden
Codex and 400 inscriptions from the cathedral cities. Using the so called
Böhm correlation coefficient we were successful in proving that all
data contained in the Dresden Codex are concerning astronomical phenomena.

The Mayan astronomical observations
were carried out by simple measuring methods. It is therefore necessary
to examine them statistically while respecting unavoidable accuracy scatter.
It applies first of all to the sky phenomena calculated to the past and
the future during several centuries recorded in the Dresden Codex. The
dated astronomical observations are concerning following:

The observations of Venus visibility,
when it had appeared for the first time after its conjunction with the
Sun as a morning star in the sky shortly before the sunrise, or after its
upper conjunction, when it had appeared in the sky as an evening star shortly
after the sunset.

The observations of Mercury visibility.
Its trajectory creates an eccentric ellipse. Thanks to this eccentricity
the synodic circulations of the planet lasts from 104 to 132 days. The
average length of the synodic circulation is 115.877484 days. The considerable
proximity of the planet to the Sun makes its glow suppressed by dazzling
sunshine. For that reason, the Mayan astronomers could have observed it
only when the planet gets to the greatest angle distance during its circulation
around the Sun, so called elongation. It is the western elongation, when
Mercury rises over the horizon shortly before sunrise and the eastern elongation,
when it is briefly visible over the western horizon right after sunset.
The maximal angle distances are as a rule moving between 18 to 23 degrees.
The maximal elongation of 27 degrees and 49 minutes happens when this elongation
visible from the Earth runs during the epihelium, that means the greatest
distance between Mercury and the Sun (Mercury gets there once for its sidereal
circulation on its eccentric trajectory). The sidereal circulation is the
real time of circulation of any planet around the Sun and it makes in this
case 87.9693 days. During the perihelia (minimal distance from the Sun)
is the maximal angle distance visible from the Earth 15 degrees and 55
minutes. It seems as if the planet stood in one place for 4-12 days in
the time of maximal elongations. Its angle distance from the Sun changes
between 1-2 degrees. This insignificant movement could not have been discovered
by the Mayan observing methods. For that reason their determination of
Mercury elongations moves on average inside the borders of that "mistake".

The solar eclipses and the fullmoons and
newmoons.

The observation of the heliacal risings
and settings of the planets. The heliacal rise sets in after the planets'
conjunction with the Sun, when they are visible in the morning sky before
the sunrise. During their heliacal setting the planets are visible in the
evening sky after sunset. In the period of conjunction the planets are
invisible for a few days. By observing the heliacal risings and settings
dates we are able to determine the length of the synodic circulations of
the planets.

The observation of the planetary conjunctions
(when two planets observable from the Earth get in line and are nearly
covering each other). The Dresden Codex is mostly describing only close
approaches of the planets because some of the dates are calculated to the
past and to the future.

The determination of equinoxes and solstices.

All the astronomical phenomena in the
Dresden Codex are counted for the visibility from the geographical area
of the Mayan culture.

The Dresden Codex is divided into pages,
that are numbered both after the older version set by E.W.Förstermann
(F) and the newer one by Ju.V.Knorozov (D). The Mayan dates covering one,
exceptionally more, pages are forming whole files. We keep their original
topological placement during rewriting, in order to get a minimal distortion
in comparison with the original. The Mayan dates in the Dresden Codex presented
in brackets are not expressed by the number of passed days, but by a calendar
circle made by combination of the dates from the 260 day tzolkin and 365
day haab. From them and from the relation to the other dates is calculated
the real value of passed days.

Each Mayan date is at first rewritten
in the Mayan calendar system, than transferred into the decadic counting
system, by means of the Böhm coefficient (622,261 days) it is recalculated
to the Julian day number (JD) and finally adjusted to the Julian calendar
system.

All the computer programs were provided
by Dr.Ing. Jan Vondrák from the Astronomical Department of the Science
Academy of Czech Republic, for which we are very thankful.The scheme of calculating the Mayan
dates to the Christian system of dating using the Böhm coefficient
of 622,261 days:

1/ The inscriptions of Mayan dates

2/ Transfers to the decimal arithmetical
system

3/ Transfers to Julian day numbers
by adding 622,261

4/ Days, months and years of the Julian
calendar

5/ Analysis of the astronomical phenomena
related to the single dates

PAGE F 24, 46-50, /D 24-29/

A/ 9.9.16.0.0 4 Ahau 8 Cumuku 1,366,560
days

- 6.2.0- 2,200 days

B/ 9.9.9.16.0 1 Ahau 18 Kayab 1,364,360
days

C/ /9.14.2.6.0/ 1 Ahau 18 Uo 1,397,640
days

A/ J.D. 1,988,821 February 4, 733

B/ J.D. 1,986,621 January 27, 727

C/ J.D. 2,019,901 March 10,818

The rounded length of Venus synodic
circulation of 584 days multiplied by five are added to the dates B and
C 5-65 times and 130; 195 and 260 times. The real length of synodic circulation
of Venus is 583.921 394 days. Time intervals of 185,120; 68,900; 33,280
and 9,100 days are added to the number B. And again the upper mentioned
multiplies of 584 days. The total of B and interval of 33.280 days is the
same as the date C, in the Dresden Codex mentioned only by the dates of
calendar circle — 1 Ahau 18 Uo.

The total of B and 68,900 or B and
9,100 days is forming next dates D and E. Their last day always falls on
1 Ahau from 260 day tzolkin, as stressed in the Codex table.

B/ 9.9.9.16.0 1 Ahau 18 Kayab 1,364,360
days

68,900 days

D/ /9.19.1.5.0/ 1 Ahau 1,433,260 days

B/ 9.9.9.16.0 1 Ahau 18 Kayab 1,364,360
days

1.5.5.09,100 days

E/ /9.10.15.3.0/ 1 Ahau 1,373,460 days

D/ J.D. 2,055,521 September 17, 915

E/ J.D. 1,995,721 December 27, 751

The rounded length of synodic circulation
of Venus multiplied by five is 2,920 days. They contain:

5 times the length of synodic circulation
of Venus

13 times the length of sidereal circulation
of Venus

8 times the length of tropical year

2,920 days is the typical period when
the sidereal and sinodical circulation of Venus meet the tropical year.
In this time the planet sets or rises in the same place in the sky after
8 years.

Venus visibility ephemerides are mentioned
in the next Codex tables. Its synodic circulation length is written like
five times /236 and 90 and 250 and 8 days/. These are comprehensible approximate
values, because of the synodic circulation length of this planet is moving
between 577 and 592 days.

236 days - Venus is in the eastern
sky as a morning star

90 days - Venus is around the upper
conjunction with the Sun and not visible

250 days - Venus is in the western
sky as an evening star

8 days - Venus is around the lower
conjunction with the Sun and not visible

This specification is added to dates
B and D five times.

Dates B, C and D are concerning the
moment of Venus first observation as a morning star after its lower conjunction
with the Sun.

Date E is concerning the moment of
Venus first observation as an evening star after tens of days of invisibility
because of its upper conjunction with the Sun.

B/ January 27, 727 Venus is 8 days
after the lower conjunction with the Sun. For first time observed as a
morning star in the eastern sky.Venus rose at 5.39 am

The Sun rose at 6.34 am

C/ March 10, 818 Venus is five days
after its lower conjunction with the Sun. For the first time observed as
a morning star in the eastern sky.Venus rose at 5.36 am

The Sun rose at 6.11 am

D/ September 17, 915 Venus is six
days after its lower conjunction with the Sun. For the first time observed
as a morning star in the eastern sky.Venus rose at 4.54 am

The Sun rose at 5.49 am

E/ December 27, 751 Venus is 63 days
after upper conjunction with the Sun. For the first time observed as an
evening star in the western sky.The Sun set at 5.42 p.m.

Venus set at 6.46 p.m.

There is a time interval of 2,200 days
marked between dates A and B. It contains:19 times Mercury synodic circulation
length /115.877 484 days/

25 times Mercury sidereal circulation
length /87.968 581 days/

6 times the tropical year /365.242
199 days/

2,200 days is the typical period when
the sidereal and sinodical circulation of Mercury meet the tropical year.
In this time the planet sets or rises in the same place in the sky after
6 years. The similar cyclical regularity was observed by the Mayan astronomers
about Venus, with a period of 2,920 days. The dates A and B are marking
the moments when Mercury was to be found in a close approach of the maximal
western elongations from the Sun. The D date concerns the maximal eastern
elongation. Only in those positions around the maximal elongations is the
planet visible not long before the sunrise or shortly after the sunset.A/ February 4, 733 Mercury is to
be found in western elongation with angle distance 26.47 degrees from the
Sun.

Mercury rose at 4.53 am

The Sun rose at 6.32 am

The maximal western elongation of 26.73
degrees was on January 31, 733. The difference between determined and real
elongation is 0.26 degrees.

B/ January 27, 727 Mercury was in western
elongation with angle distance 25.5 degrees from the Sun.

Mercury rose at 4.55 am

The Sun rose at 6.34 am

The maximal western elongation of 26.07
degrees was on January 21, 727. The difference between determined and real
elongation is 0.57 degrees.

D/ September 17, 915 Mercury was in
eastern elongation with angle distance 25.05 degrees from the Sun.

In all three determined elongations the
Mayan astronomers have made an average mistake 0.3 degrees less than reality.
It is so slight difference that it can't have been discovered.E/ December 27, 751 The conjunction
of Venus, Mercury and Jupiter. The position of planets is expressed in
degrees of geocentrical ecliptical coordinates:

Mercury 295.489

Venus 294.289

Jupiter 294.987

The time interval 2,200 days between A
and B dates contains 74.5 times the length of synodic circulation of the
Moon /29.530 588 days/.

A/ February 4, 733 The full moon

B/ January 27, 727 The new moon

PAGE F 51 - 58, /D 30 - 37/

A/ 8.16.4./8/.0 4 Ahau 1,268,800 days

88 days

B/ 8.16.4./8/.8 12 Lamat 1,268,808
days

C/ 10.19.6./1/.8 12 Lamat 1,578,988
days

D/ 9.19./5/.7.8 7 Lamat 1,434,748 days

E/ 9.16.4./11/.18 3 Etznab 1,412,878
days

F/ 9.16.4.11.3 1 Akbal 1,412,863 days

G/ 9.16.4.10.8 12 Lamat 1,412,848 days

A/ J.D. 1,891,061 June 11, 465

B/ J.D. 1,891,069 June 19, 465

C/ J.D. 2,201,249 September 10, 1314

D/ J.D. 2,057,009 October 14, 919

E/ J.D. 2,035,139 November 28, 859

F/ J.D. 2,035,124 November 13, 859

G/ J.D. 2,035,109 October 29, 859

The multiplies of 11,960 days (2,3,4,5,16,17,18
and 39 times) and of 13,780; 71,880 and 371,020 days are added to dates
B, C, D, E, F and G.

B/ June 19, 465 The summer solstice

There is an interval of 144,240 days
between C and D dates. It contains (with little differences of few days):

247 times Venus synodic circulation
length /583.921 394 days/

642 times Venus sidereal circulation
length /224.700 800 days/

185 times Mars synodic circulation
length /779.936 160 days/

210 times Mars sidereal circulation
length /686.979 800 days/

394.916 tropical years

The period of 394.916 tropical years
contains the basic cycles of Mars and Venus conjunctions:

305.352 years the first conjunction
cycle

14 times 6.3974 years the second conjunction
cycle

C/ September 10, 1314 Mars and Venus
conjunction.

Mars 197.76 degrees.

Venus 187.99 degrees.

D/ October 14, 919 Mars and Venus conjunction.

Mars 214 degrees.

Venus 215.48 degrees.

The position of both the planets
is expressed in degrees of geocentrical ecliptical co-ordinates.

The close approach (1.5 degrees) of the
planets was visible, so the D date could be considered as an actual one.
The C date is calculated 395 years to future. Therefore the mistake of
9.77 degrees occurred. The Mayan astronomers have been calculating only
with the approximate values of the planets synodic circulation length.
The approximate values make difference of several days in the end.

E/ November 28, 859 The new moon.

F/ November 13, 859 The full moon.

G/ October 29, 859 Annular solar eclipse.

Beginning at 2.28 p.m. ephemeride time

Maximum at 5.19 p.m. ephemeride time

End 8.10 p.m. ephemeride time

The eclipse maximum was visible at
11.19 a.m. local time in the Mayan area centre (90 degrees western length;
16 degrees northern width).

The multiple of 11,960 days is added
to date G. It is the interval of solar eclipses counted by the Mayan astronomers.
It is divided into shorter intervals of 148 and 177 and 178 days following
in the tables in certain system. They contain the gathering of the Moon
synodic circulation length /29.530 588 days/ and draconic circulation length
/27.212 219 days/, which is the time, when the Moon on its way around the
Earth crosses the ecliptic twice, this means a half of the draconic circulation
length. In newmoon the solar eclipse sets, in fullmoon the moon eclipse
sets. In 11,960 days, there are 405 synodic and 439.5 draconic circulations
contained.

Adding shorter intervals of 148 and
177 and 178 days to the starting date G, we get dates concerning the Sun
again. But after a longer period the mistake grows to 1 or 2 days or more.
The Mayan astronomers were aware of that problem, so each theoretically
counted number (in the tables) could be additionally enlarged (1 or 2 days)
in frame of 260-days tzolkin. Comprehensibly, not each counted solar eclipse
was visible from the Mayan region. 73 solar eclipses visible from different
places on Earth occurred in 11,960 days starting from day G.

Due to adding intervals of 148 and
177 and 178 days, in some cases, the Mayan astronomers made such a mistake
that no eclipse occurred. In spite of that they were successful in determining
63 solar eclipses very well visible in the Mayan area during the basic
cycle of 11,960 days. So their method was quite right — adding shorter
intervals to the starting date G and creating tables of all seven expected
eclipses. Their course, i.e. the beginning, maximum and the end, is expressed
in ephemeride time. The maximum is then expressed in the local time of
approximate center of the Mayan area — 90° western longitude and 16°
northern latitude.

April 15,869 Annular solar eclipse.

The beginning at 4.47 p.m.

The maximum at 7.44 p.m.

The end at 10.40 p.m.

The maximum in the centre of the
Mayan area at 1.44 p.m.April 4, 870 Annular solar eclipse.

The beginning at 6.17 p.m.

The maximum at 9.22 p.m.

The end at 0.27 a.m.

The maximum in the centre of the
Mayan area at 3.22 p.m.May 16, 877 Total solar eclipse.

The beginning at 4.48 p.m.

The maximum at 7.34 p.m.

The end at 10.21 p.m.

The maximum in the centre of the
Mayan area at 1.34 p.m.March 14, 880 Annular solar eclipse.

The beginning at 3.32 p.m.

The maximum at 6.28 p.m.

The end at 9.24 p.m.

The maximum in the centre of the
Mayan area at 12.28.August 28, 881 Annular solar eclipse.

The beginning at 11.49 a.m.

The maximum at 2.48 p.m.

The end at 5.48 p.m.

The maximum in the centre of the
Mayan area at 8.48 a.m.June 26, 884 Total solar eclipse.

The beginning at 3.19 p.m.

The maximum at 6.02 p.m.

The end at 8.46 p.m.

The maximum in the centre of the
Mayan area at 12.02.July 27, 892 Annular solar eclipse.

The beginning at 11.44 a.m.

The maximum at 2.35 p.m.

The end at 5.26 p.m.

The maximum in the centre of the
Mayan area at 8.35 a.m.Precisely the Mayan eclipse cycle of 11,960
days is contained between October 29, 859 (the starting date G) and July
27, 892 (the last date). It could be compared to the Babylonian Saros of
6585.3 according to which the eclipses were foretold (since 6.century B.C.).

PAGE F 58 - 59 /D 37 - 38/

A/ 9.18.2.2.0 4 Ahau 1,426,360 days

- 12.11- 251 days

B/ /9.18.1.7.9/ 13 Muluc 1,426,109
days

C/ 9.12.11.11.0 4 Ahau 1,386,580 days

- 1.7.11- 511 days

D/ /9.12.10.3.9/ 13 Muluc 1,386,069
days

A/ J.D. 2,048,621 October 26, 896

B/ J.D. 2,048,370 February 18, 896

C/ J.D. 2,008,841 November 28, 787

D/ J.D. 2,008,330 July 5, 786

The multiples of (1 to 9)x78, (1 to
18)x780 and (1 to 9)x14,820 days are added to dates B and D. Any multiple
of 780 could be reached by mere addition through their mutual combination.

The interval of 39,780 days is contained
between dates A and C. It contains with little differences of just a few
days:

Venus synodic circulation length is
moving between 577 to 592 days. Its real average length is 583.921 394
days. Seemingly irregular sling movement of the planet 542 days forwards
and 42 days backwards arises from adding the movements of the Earth to
the movements of Venus. Nearly the same movement is performed by Mars.
Mars synodic movement length is moving between 764 and 810 days. The direct
movement is about 706 days and the reverse one about 74 days. Mars average
synodic circulation length is 779.936 160 days. The basic theoretical cycle
of Venus and Mars conjunction cycle is 2,340 days. And again, it moves
between 2,315 and 2,360 days.

The Mayan astronomers were obviously
aware of Venus and Mars mutual complicated movements, so they have been
working with the rounded values, which enabled them determining the planets
conjunctions for a long period with little mistake. The position of Venus
and Mars is expressed in degrees of geocentric ecliptic co-ordinates.

A/ October 26, 896 Venus and Mars conjunction.

Venus 176.267°

Mars 175.374°

The distance of the planets was 0.893°

The real conjunction held 2 days ago
— October 24,896

Venus 173.916°

Mars 174.117°

The distance of both the planets was
0.201°.

C/ November 28, 787 Venus and Mars
conjunction.

Venus 207.547°Mars 204.765°

The distance of the planets was 2.782°

The real conjunction held 1 day ago
— November 27, 787

Venus 206.870°

Mars 204.115°

The distance of both the planets was
2.755°.

In those cases the Mayan astronomers
have determined Venus and Mars conjunction a day or two later. The angle
distance has changed only slightly during this period, so it was not observable.
Their mutual shift of distance was only 0.692° and 0.027°.

If the conjunction is held during Venus
being as a morning star close to the eastern elongation with the Sun, then
the next conjunction is held around the western elongation, when Venus
shines as a morning star in the sky. The interval between two following
conjunctions is during those constellations very short and it moves between
206 and 295 days. The average value of this scatter is 251 days and it
is to be found between dates A and B.

The A date introduces Venus and Mars
conjunction. At the same time Venus was after western elongation with the
Sun. Since this date, the time interval of 241 days (average time from
the last conjunction — date B) is counted, till Venus was shortly before
eastern elongation with the Sun.

B/ February 18, 896 Venus and Mars
conjunction.

Venus 10.896°

Mars 9.966°

The distance of the planets was 0.93°

The real conjunction held 2 days ago
— February 16,896

Venus 8.504°

Mars 8.507°

The distance of both the planets was
0.003°.

In this case, the Mayan astronomers
have made a mistake of 0.897°, of which the planets receded since the
real conjunction.

In all three determined conjunctions
the Mayan astronomers have made a mistake moving from 0.027°to 0.897°,
of which the planets receded since the real conjunction. It is so slight
difference, that the planets movement on ecliptics in such small angle
distance was undiscoverable for the Mayan astronomers.

Mars and Jupiter conjunctions are in
average repeating after 2.21 years. They are together around their oppositions
with the Sun once in 49.14 years. It is the time of the best conditions
for their observation. Repetition of triple conjunctions or close mutual
approaches occurs during a short time period (tens of days) while the planets
in this position.

The D date, July 5, 786 expresses Mars
and Jupiter short-period maximal distance between two following conjunctions.
At the same time the planets were close to their oppositions with the Sun.

The first maximal approach — May 4,
786.

Mars 256.336°

Jupiter 267.078°

The distance of both the planets was
10.742°.

D/ July 5, 786 The maximal angle distance
after the first approach.

Mars 243.621°

Jupiter 260.162°

The distance of both the planets was
16.541°

The second conjunction — August 22,
786.

Mars 258.101°

Jupiter 258.314°

The distance of both the planets was
0.213°.

The Mayan astronomers have been observing
the mutual conjunctions of the outer planets (Mars, Jupiter, Saturn) during
their oppositions with the Sun. In these positions were the planets excellently
observable. We can say the same of dating Jupiter and Saturn conjunction
on page F 45 (D 74).

The table of 780 days multiples follows
after the opening Mayan dates. It is slightly rounded length of Mars synodic
circulation (779.936 16 days). Through the mutual combination of those
multiples we can get any multiple of 780 days. The table serves to determining
the next conjunctions, or close approaches, of Mars and Venus, or Mars
and Jupiter.

Mars and Venus conjunction is repeating
after three synodic circulations of Mars, i.e. 2.340 days. By mere adding
this cycle to the B date (February 18, 896), we get the time of the next
conjunction.

The mistake has increased and the planets
receded more and more after every cycle because of the Mayan astronomers
counting with the rounded values. In spite of that the method was usable
for 185.795 years with 29 basic cycles of conjunctions. The angle distance
of both the planets was after this time only 7.685°. But for calculation
of the next conjunctions is the triple multiple of 780 days no longer usable.

For calculating the next conjunctions
of Mars and Venus, it is necessary to change for another cycle. 140 synodic
circulations of Mars are again added to the starting Mayan date B. 140
synodic circulations of Mars are 109,200 days, i.e. nearly 299 years, with
187 synodic circulations of Venus. The planets get into the near approach
after this time again. For a certain time, it is possible to add a shorter
basic interval of 2,340 days (3 times 780 days) to this conjunction to
get their conjunctions or close approaches, before the mistake grow shows
again. Out of the table of 780 days multiplies we can easily find out the
multiply of 140 and 780 days and the total of 103,740 and 5,460 days is
109,200 days.

The parallel positions of Mars and
Jupiter nearby their oppositions with the Sun (when they also get into
mutual conjunction) are repeating every 49.14 years with 23 synodic circulations
of Mars, i.e. 17,940 days. Out of the table of 780 days multiplies we can
find this value by adding 14,820 days to 3,120 days, which is exactly 17,940
days. By gradual adding the multiplies of this value to the initial Mayan
date D (July 5, 786), Mars and Jupiter are always reaching the opposition
with the Sun, when they are good observable, but they are also situated
approximately between two following conjunctions. Because of working with
average values of astronomical data, the mistake grows approximately after
147.4 years.

PAGE F 61 - 64 /D 40 - 43/

A/ /?/.15.9.1.3 The Mayan date value
is undeterminable.

B/ 9 Kan 12 Kayab A cyclic date repeating
every 18,980 days.

The Mayan date final value in the chronology
system is undeterminable.

C/ 8.16.15.16.1 1,272,921 days

- 1.4.16- 456 days

D/ /8.16.14.11.5/ 3 Chicchan 1,272,465
days

E/ 8.16.14.15.4 1,272,544 days

- 6.1- 121 days

F/ /8.16.14.9.3/ 13 Akbal 1,272,423
days

G/ 8.11.8.7.0 1,234,220 days

- 11.15- 235 days

H/ /8.11.7.13.5/ 3 Chicchan 1,233,985
days

I/ 8.16.3.13.0 1,268,540 days

- 0.17- 17 days

J/ /8.16.3.12.3/ 13 Akbal 1,268,523
days

K/ 10.8.3.16.4 1,499,004 days

L/ 10.13.13.3.2 1,538,342 days

- 7.2.14.19- 51,419 days

M/ /10.6.10.6.3/ 13 Akbal 1,486,923
days

N/ /8/.19.0.4.4 1,288,884 days

C/ J.D. 1,895,182 September 22, 476

D/ J.D. 1,894,726 June 24, 475

E/ J.D. 1,894,805 September 11, 475

F/ J.D. 1,894,684 May 13, 475

G/ J.D. 1,856,481 October 8, 370

H/ J.D. 1,856,246 February 15, 370

I/ J.D. 1,890,801 September 24, 464

J/ J.D. 1,890,784 September 7, 464

K/ J.D. 2,121,265 September 16, 1095

L/ J.D. 2,160,603 May 30, 1203

M/ J.D. 2,109,184 August 19, 1062

N/ J.D. 1,911,145 June 6, 520

Behind the set of the Mayan dates follow
the tables containing multiplies of (1 to 20)x91 days, 15x364, 20x364,
40x364, 40x364, 60x364 and 80x364 days. The record of higher dates is damaged.

The Mayan dates C, E, G, I and N are
concerning Jupiter heliacal sets, i.e. the moments of the last short visibility
over the western horizon shortly after the sunset. The planet was invisible
for the next few tens of days because of its conjunction with the Sun.
Jupiter synodic circulation length multiplies are contained between the
dates E, G, I and N. It moves between 395 and 404 days. The average is
398.884 070 days.

E/ September 11, 475

38,324 days = 96 Jupiter synodic circulation
length

G/ October 8, 370

34,320 days = 86 Jupiter synodic circulation
length

I/ September 24, 464

20,344 days = 51 Jupiter synodic circulation
length

N/ June 6, 520

C/ September 22, 476 Jupiter is 41
days before conjunction with the Sun.

The heliacal set of the planet.

The Sun set at 5.56 p.m.

Jupiter set at 7.47 p.m.

The autumn equinox.

E/ September 11, 475 Jupiter is 23
days before conjunction with the Sun.

The heliacal set of the planet.

The Sun set at 6.05 p.m.

Jupiter set at 7.06 p.m.

11 days before the autumn equinox.

G/ October 8, 370 Jupiter is 49 days
before conjunction with the Sun.

The heliacal set of the planet.

The Sun set at 5.45 p.m.

Jupiter set at 8.03 p.m.

16 days after the autumn equinox.

I/ September 24, 464 Jupiter is 35
days before conjunction with the Sun.

The heliacal set of the planet.

The Sun set at 5.54 p.m.

Jupiter set at 7.29 p.m.

2 days after the autumn equinox.

N/ June 6, 520 Jupiter is 44 days before
conjunction with the Sun.

The heliacal set of the planet.

The Sun set at 6.27 p.m.

Jupiter set at 8.50 p.m.

14 days before the summer solstice.

The results of observed, or into past
calculated heliacal sets of Jupiter can be summarized into following conclusion:

1/ The heliacal sets set in on average
39 days before conjunction with the Sun. This corresponds to the time,
when the planet is for the last time shortly visible after the sunset over
the western horizon, before it gets into conjunction with the Sun.

2/ Jupiter heliacal sets always happened
approximately in the prominent parts of the tropical year. Four times around
the autumn equinox and once around the summer solstice.

3/ The Sun set in a very short time
interval of 20 minutes, since 5.45 till 6.05 p.m. in cases C, E, G and
I. Jupiter set some 57 minutes, since 7.06 till 8.03 p.m.

The Mayan dates C, E, H, L and M are
concerning Saturn heliacal rises, i.e. the moments when the planet was
visible for the first time in the morning sky before the sunrise. It was
unobservable for few tens of days before, because of its conjunction with
the Sun. Saturn synodic circulation length multiples are contained in all
these data. With a slight scatter the circulation makes 378.091 900 days.

C/ September 22, 476

377 days = 1x Saturn synodic circulation
length

E/ September 11, 475

38,559 days = 102x Saturn synodic circulation
length

H/ February 15, 370

304,357 days = 805x Saturn synodic
circulation length

L/ May 30, 1203

51,419 days = 136x Saturn synodic circulation
length

M/ August 19, 1062

C/ September 22, 476 Saturn is 43
days after conjunction with the Sun.

The heliacal rise of the planet.

Saturn rose at 3.23 a.m.

The Sun rose at 5.49 a.m.

E/ September 11, 475 Saturn is 45 days
after conjunction with the Sun.

The heliacal rise of the planet.

Saturn rose at 3.13 a.m.

The Sun rose at 5.47 a.m.

H/ February 15, 370 Saturn is 63 days
after conjunction with the Sun.

The heliacal rise of the planet.

Saturn rose at 2.53 a.m.

The Sun rose at 6.29 a.m.

L/ May 30, 1203 Saturn is 61 days after
conjunction with the Sun.

The heliacal rise of the planet.

Saturn rose at 2.30 a.m.

The Sun rose at 5.26 a.m.

M/ August 19, 1062 Saturn is 51 days
after conjunction with the Sun.

The heliacal rise of the planet.

Saturn rose at 2.44 a.m.

The Sun rose at 5.45 a.m.

Two files of following Mayan dates are
concerning Mercury positions close to its maximal western or eastern elongations.
The G, J, C, K, H and D dates are concerning the western elongations, when
the planet was rising in the eastern sky before the sunrise. The F, N and
E dates are concerning the eastern elongations. Mercury set in the evening
sky after the sunset. The planet was practically observable only in those
positions, when it got into the maximal angle distances from the Sun. Mercury
synodic circulation length multiples are contained between the dates, incidentally
the synodic and sidereal circulations and the approximate tropical year
conformities.

G/ October 8, 370

34,303 days = 296x Mercury synodic
circulation length

390x Mercury sidereal circulation length

94x the tropical year length

J/ September 7, 464

4, 398 days = 38x Mercury synodic circulation
length

50x Mercury sidereal circulation length

12x the tropical year length

C/ September 22, 476

226,083 days = 1,951x Mercury synodic
circulation length

2,570x Mercury sidereal circulation
length

619x the tropical year length

K/ September 16, 1095

265,019 days = 2,287x Mercury synodic
circulation length

H/ February 15, 370

38,480 days = 332x Mercury synodic
circulation length

D/ June 24, 475

F/ May 13, 475

16,461 days = 142x Mercury synodic
circulation length

187x Mercury sidereal circulation length

45x the tropical year length

N/ June 6, 520

16,340 days = 141x Mercury synodic
circulation length

E/ September 11, 475

If Mercury synodic and sidereal circulation
length meets the tropical year approximate length, the maximal elongations
of the planet are repeating in positions concerning (with little differences)
the conformity of:

1/ the planet rises and sets time,

2/ the Sun rises and sets time,

3/ the approximate position of the
planet on its ecliptics,

4/ the date of the year with difference
smaller than 30 days.

G/ October 8, 370 Mercury is close
to the western elongation with angle distance 17.61° from the Sun.

Mercury rose at 4.44 a.m.

The Sun rose at 5.52 a.m.

The real maximal elongation of 18.83°
was on October 12, 370.

The difference between the estimated
and real elongation is 1.22°.

J/ September 7, 464 Mercury is close
to the western elongation with angle distance 17.27° from the Sun.

Mercury rose at 4.37 a.m.

The Sun rose at 5.46 a.m.

The real maximal elongation of 17.92°
was on September 10, 464.

The difference between the estimated
and real elongation is 0.65°.

C/ September 22, 476 Mercury is close
to the western elongation with angle distance 14.51° from the Sun.

Mercury rose at 4.55 a.m.

The Sun rose at 5.49 a.m.

The real maximal elongation of 18.33°
was on September 29, 476.

The difference between the estimated
and real elongation is 3.82°.

K/ September 16, 1095 Mercury is close
to the western elongation with angle distance 17.61° from the Sun.

Mercury rose at 4.39 a.m.

The Sun rose at 5.49 a.m.

The real maximal elongation of 17.94°
was on September 18, 1095.

The difference between the estimated
and real elongation is 0.33°.

H/ February 15, 370 Mercury is close
to the western elongation with angle distance 26.08° from the Sun.

Mercury rose at 4.58 a.m.

The Sun rose at 6.30 a.m.

The real maximal elongation of 27.69°
was on February 22, 370.

The difference between the estimated
and real elongation is 1.61°.

D/ June 24, 475 Mercury is close to
the western elongation with angle distance 19.25° from the Sun.

Mercury rose at 4.11 a.m.

The Sun rose at 5.23 a.m.

The real maximal elongation of 19.96°
was on June 27, 475.

The difference between the estimated
and real elongation is 0.71°.

The Mayan dates are determining Mercury
western elongations on average four days sooner, before the real moment
of the maximal elongation. The Mayan astronomers have made an average mistake
1.39°, which was not measurable at all because of the difficulties
connected with observing this planet. The biggest departure of -3.82°
can be found on September 22, 476 — the date C. Mercury synodic and sidereal
circulation length and the approximate tropical year meets between the
dates G, J, C and K. The determined elongations are probably results of
theoretical calculations. The date K into the future, the next dates into
the deep past. Therefore a bigger mistake occurred by the C date, if only
average values of the planet circulation length were used. Mercury synodic
circulation length vary from 104 to 132 days.F/ May 13, 475 Mercury is close to
the eastern elongation with angle distance 23.41° from the Sun.

The Sun set at 6.22 p.m.

Mercury set at 8.07 p.m.

The real maximal elongation of 23.51°
was on May 10, 475.

The difference between the estimated
and real elongation is 0.1°.

N/ June 6, 520 Mercury is close to
the eastern elongation with angle distance 23.49° from the Sun.

The Sun set at 6.30 p.m.

Mercury set at 8.10 p.m.

The real maximal elongation of 25.15°
was on May 25, 520.

The difference between the estimated
and real elongation is 1.66°.

E/ September 11, 475 Mercury is close
to the eastern elongation with angle distance 25.21° from the Sun.

The Sun set at 6.05 p.m.

Mercury set at 7.16 p.m.

The real maximal elongation of 25.70°
was on September 6, 475.

The difference between the estimated
and real elongation is 0.49°.

The Mayan astronomers determined the eastern
elongation on average 5 days later, after the planets maximal distance
from the Sun. They have made an average mistake 0.75°, which was due
to their methods undiscoverable. The angle distance of Mercury and the
Sun changes only slightly for a few days around the maximal elongations.
The planet seemingly stands at one place. Therefore the Mayan astronomers
could not have reached bigger accuracy.

The following time intervals a to h
are added to the cyclic date B /9 Kan 12 Kayab/. We can not determine the
real value of this date in the Mayan chronology system and therefore recalculate
it into the Christian dating system.

a/ 4.6./1./11.3.1. 3 Chicchan 12,395,221
days

b/ 4.6./13./13.15.1 3 Chicchan 12,482,581
days

c/ 4.6.1.9.15.0 3 Kan 12,394,740 days

d/ 4.6.9.16.10.1 3 Chicchan 12,454,761
days

e/ 4.6.7.12.4.10 3 Ix 12,438,810 days

f/ 4.6.11.10.7.2 3 Cimi 12,466,942
days

g/ 4.6.9.15.12.19 13 Akbal 12,454,459
days

h/ 4.6.1.9.15.0 3 Kan 12,394,740 days

Basing on the analysis of single time
sections and mutual intervals between them we found out, that they are
concerning the observations of tropical year for very long time period,
but it is impossible to match them with Christian calendar dates.

a/ 12,395,221 days 33,937 tropical
years with +3 days departure

d/ 12,454,761 days 34,100 tropical
years with -3 days departure

The interval between dates e - f contains
77 tropical years with -8 days departure.

The interval between dates g - b contains
77 tropical years with +2 days departure.

The interval between dates c - b contains
240.5 tropical years.

The interval between dates c - g contains
163.5 tropical years with -2 days departure.

The intervals between some time sections
also contain some planets synodic circulations, what theoretically correspondences
with their conjunctions.

Time interval of 87,360 days between
dates a-b = Mars, Jupiter and Saturn conjunction.

Time interval of 72,202 days between
dates c-f = Jupiter and Saturn conjunction.

Time interval of 12,483 days between
dates f-g = Mars and Saturn conjunction.

The Mayan dates E, F, I, J, L and M
are repeated on page F 31-32 /D 60-61/. Than follows the table of 91 and
364 multiplies, which are added to the dates.

All the visible planets were rising
or setting close to the maximal northern declination of the Sun, i.e. in
the place, where the Sun rises and sets during the summer solstice.

Declinations: Sun 23.40°

Mercury 24.41°

Venus 18.54°

Mars 23.56°

Jupiter 23.47°

Saturn 23.32°

The Mayan dates C, N and M are concerning
the dates of Venus last visibility in the western sky before its lower
conjunction with the Sun. Venus synodic circulation lengths (583.92139
days) are contained between those dates.

C/ July 20, 808

13,434 days = 23 Venus synodic circulation
lengths.

N/ May 1, 845

134,298 days = 230 Venus synodic circulation
lengths.

M/ January 7, 1213

C/ July 20, 808 Venus 8 days before
the lower conjunction with the Sun.

Seen for the last time as an evening
star in the western sky.

The Sun set at 6.32 p.m.

Venus set at 7.06 p.m.

N/ May 1, 845 Venus 6 days before the
lower conjunction with the Sun.

Seen for the last time as an evening
star in the western sky.

The Sun set at 6.18 p.m.

Venus set at 7.09 p.m.

M/ January 7, 1213 Venus 12 days before
the lower conjunction with the Sun.

The intervals between Mercury positions
are only approximate and can differ a few days after every finished synodic
circulation with length varying between 104 to 132 days. This is caused
by a great eccentricity of Mercury trajectory around the Sun.

The Mayan dates are recording Mercury
positions close to the four basic positions that occur during its circulation
around the Sun. They can be divided into four files:

The maximal elongations are observable
when the planet sets in the longest time interval after the sunset during
the eastern elongation, or rises before the sunrise during the western
elongation. The lower and upper conjunction (the planet is unobservable)
was probably counted by the Mayan astronomers as the middle of time intervals
between two following maximal elongations. In the tables concerning Venus
visibility they also counted 8 and 90 days, when the planet was unobservable,
because it was around the lower or upper conjunction with the Sun — pages
F 24, 46-50, /D 24-29/.

Mercury synodic circulation lengths
are contained between all the dates in each file. Between some multiplies
of synodic and sidereal circulation meetings (87.9693 days) with approximate
tropical year length (365.242 199 days). That means approximately same
mutual positions of Mercury, Sun and Earth were held after some time.

File 1 — the maximal eastern elongation
- dates L,D and G

L/ October 28, 479

119,462 days = 1,031 Mercury synodic
circulation lengths.

1,358 Mercury sidereal circulation
lengths.

327 tropical year lengths.

D/ November 22, 806

173,818 days = 1,500 Mercury synodic
circulation lengths.

1,976 Mercury sidereal circulation
lengths.

476 tropical year lengths.

G/ October 12, 1282

L/ October28, 479 Mercury is close
to the eastern elongation with angle distance 21.34° from the Sun.

The Sun set at 5.34 p.m.

Mercury set at 6.46 p.m.

The real maximal elongation of 22.40°
was on October 22, 479.

The difference between the estimated
and real elongation is 1.06°.

D/ November 22, 806 Mercury is close
to the eastern elongation with angle distance 20.21° from the Sun.

The Sun set at 5.28 p.m.

Mercury set at 6.48 p.m.

The real maximal elongation of 20.30°
was on November 23, 806.

The difference between the estimated
and real elongation is 0.09°.

G/ October 12, 1282 Mercury is in the
eastern elongation with angle distance 23.61° from the Sun.

The Sun set at 5.37 p.m.

Mercury set at 6.52 p.m.

The Mayan astronomers have made an average
mistake of 0.38° in those three elongations — for them an undiscoverable
mistake.

File 2 — the maximal western elongation
- dates I and E

I/ May 13, 280

235,820 days = 2,035 Mercury synodic
circulation lengths.

E/ January 2, 926

I/ May 13, 280 Mercury is close to
the western elongation with angle distance 23.06° from the Sun.

Mercury rose at 4.19 a.m.

The Sun rose at 5.29 a.m.

The real maximal elongation of 23.33°
was on May 15, 280.

The difference between the estimated
and real elongation is 0.27°.

E/ January 2, 926 Mercury is close
to the western elongation with angle distance 21.54° from the Sun.

Mercury rose at 5 a.m.

The Sun rose at 6.33 a.m.

The real maximal elongation of 23.55°
was on

December 22, 925.

The difference between the estimated
and real elongation is 2.01°.

The Mayan astronomers have made an average
mistake of 1.14° in those two elongations — for them an undiscoverable
mistake.

The dates are concerning Mercury positions
around the lower conjunction with the Sun. The Mayan astronomers have probably
determined this position with help of the visible eastern and western elongations,
when Mercury is best visible. They set an approximate middle of the time
Mercury was unobservable (between two following maximal elongations and
around the lower conjunction with the Sun).

C/ July 20, 808 Mercury 2 days before
the lower conjunction.

26 days after the eastern elongation.

19 days before the western elongation.

F/ July 1, 921 Mercury 1 day after
the lower conjunction.

29 days after the eastern elongation.

18 days before the western elongation.

P/ January 2, 635 Mercury 7 days after
the lower conjunction.

23 days after the eastern elongation.

17 days before the western elongation.

H/ December 25, 1154 Mercury 5 days
after the lower conjunction.

21 days after the eastern elongation.

19 days before the western elongation.

M/ January 7, 1213 Mercury 1 day before
the lower conjunction.

14 days after the eastern elongation.

26 days before the western elongation.

N/ May 1, 845 Mercury 1 day before
the lower conjunction.

22 days after the eastern elongation.

26 days before the western elongation.

The results of the Mayan astronomers
calculations could be using a statistical average method simplified into
following conclusion, that is concerning Mercury position close to the
lower conjunction with the Sun:

1/ 1.3 days before the lower conjunction.

2/ 4.3 days after the lower conjunction.

3/ 22.5 days after the eastern elongation.

4/ 20.8 days before the western elongation.

The average length of time to the dated
position of Mercury after eastern elongation and before western elongation
is 21.7 days. This precisely corresponds with the real average time length
from position - eastern elongation to lower conjunction and from lower
conjunction to western elongation, which lasts for 22 days on average,
as shown before.

File 4 — the upper conjunction with
the Sun — dates J and K.

J/ February 17, 280

73.126 days = 631 Mercury synodic circulation
lengths.

K/ May 3, 480

The dates are concerning Mercury positions
around the upper conjunction with the Sun. In a similar way as with the
lower conjunction, the Mayan astronomers have set an approximate middle
of the time Mercury was unobservable between both the maximal elongations
and around the upper conjunction with the Sun.

J/ February 17, 280 Mercury 14 day
before the upper conjunction.

31 days after the western elongation.

40 days before the eastern elongation.

K/ May 3, 480 Mercury 1 day before
the upper conjunction.

38 days after the western elongation.

36 days before the eastern elongation.

We can gather the results of dated
positions of Mercury around lower conjunction with the Sun in a statistical
average:

1/ 7.5 days before the upper conjunction.

2/ 34.5 days after the western elongation.

3/ 38 days before the eastern elongation.

The time lengths average, when Mercury
was accordingly to the Mayan dating in position after the western elongation
and before eastern elongation, is 36.25 days. This precisely correspondences
with the real time length average from western elongation to upper conjunction
and from upper conjunction to eastern elongation, which is on average 36
days, as shown before.

Following time interval a is added
to the cyclical date B and time interval b is probably added to date A.
Because we can not reliably determine the value of dates A-B in the Mayan
calendar system, it is impossible to match their total with time intervals
to particular dates of the Christian calendar.

a/ 4.5.19.13.12.8 4 Eb 12,381,728 days

b/ 4.6.19.0./12./ 10 9 Ix 12,521,050
days

The time intervals are probably concerning
the tropical year observations for very long time intervals.

The planet was in position, when it
is best observable around the opposition with the Sun. The declination
was 19.33°. This means, it was going through the centre of Mayan area
at 10.42 p.m. nearly in zenith. It set at 5.12 a.m. The Sun rose at 6.18
a.m.

B/ March 12, 922 Mars is 59 days after
conjunction with the Sun.

The heliacal rise of the planet.

Mars rose at 5.32 a.m.

The Sun rose at 6.08 a.m.

During the heliacal rise Mars was for
the first time shortly visible in the morning sky shortly before the sunrise.
It was unobservable for few tens of days, because it was going through
the area of conjunction with the Sun.

PAGE F 45 / D 74/

A/ 8.17.11.3.0 4 Ahau 1,278,420 days

- 1.10- 30 days

B/ /8.17.11.1.10/ 13 Oc 1,278,390 days

A/ J.D. 1,900,681 October 13, 491

B/ J.D. 1,900,651 September 13, 491

(2-5)x364, 10x364, 15x364, 40x364?,
60x364? and 80x364 days are added to date B.

A/ October 13, 491 Mars is 26 days
before opposition with the Sun.

The planet was in position, when it
is best observable around the opposition with the Sun. The declination
was 19.05°, so quite same as by the date A (February 27, 923) on page
F 43-44 /D 72-73/. This means, it was going through the approximate centre
of Mayan area at 2.11 a.m. nearly in zenith.

B/ September 13, 491 Jupiter and Saturn
conjunction.

Jupiter 307.827°.

Saturn 306.157°.

The distance between the planets 1.67°.

Jupiter was 41 days and Saturn 44 days
after the opposition with the Sun. So they were in positions, when the
best visible.

The real conjunction was held on October
2, 491.

Jupiter 307.221°.

Saturn 305.645°.

The distance between the planets 1.576°.

The Mayan astronomers have made a mistake
of only 0.094°, which was undiscoverable.

In the 364 days multiplies table, added
to date B (September 13, 491), are as a final value marked 80x364 days,
which are 29.120 days. They contain:

73 Jupiter synodic circulation lengths.

77 Saturn synodic circulation lengths.

It is the period with 19.87 +- 0.8
years four times contained, which is the basic interval of Jupiter and
Saturn conjunctions repeating. By adding 29,120 days to the starting date
B (September 13, 491) marking a conjunction of both the planets, we get
a new date — June 5, 571, when the planets got to a close approach again,
with distance 4.081°near to their oppositions with the Sun. Jupiter
46 and Saturn 43 days after the opposition.